RESOURCE AWARE DISTRIBUTED DETECTION AND ESTIMATION OF RANDOM EVENTS
IN WIRELESS SENSOR NETWORKS
by
ENG˙IN MAS¸AZADE
Submitted to the Graduate School of Engineering and Natural Sciences in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
Sabancı University June 2010
RESOURCE AWARE DISTRIBUTED DETECTION AND ESTIMATION OF RANDOM EVENTS
IN WIRELESS SENSOR NETWORKS by Engin Ma¸sazade
APPROVED BY
Assoc. Prof. Dr. Mehmet Keskin¨oz ... (Thesis Advisor)
Prof. Dr. Pramod K. Varshney ... (Thesis Co - Advisor)
Assist. Prof. Dr. Ruixin Niu ...
Assist. Prof. Dr. G¨ull¨u Kızılta¸s S¸endur ...
Assist. Prof. Dr. M¨ujdat C¸ etin ...
c
°Engin Ma¸sazade, 2010 All Rights Reserved
RESOURCE AWARE DISTRIBUTED DETECTION AND ESTIMATION OF RANDOM EVENTS
IN WIRELESS SENSOR NETWORKS
Engin Ma¸sazade
PhD Thesis, 2010
Thesis Advisor: Assoc. Prof. Dr. Mehmet Keskin¨oz Thesis Co - advisor: Prof. Dr. Pramod K. Varshney
Keywords: Distributed detection, distributed estimation, multi-objective optimization, source localization, sensor selection, wireless sensor net-works, wireless communication
In this dissertation, we develop several resource aware approaches for detection and estimation in wireless sensor networks (WSNs). Tolerating an acceptable degrada-tion from the best achievable performance, we seek more resource efficient soludegrada-tions than the state-of-the-art methods. We first define a multi-objective optimization problem and find the trade-off solutions between two conflicting objectives for the distributed detection problem in WSNs: minimizing the probability of error and minimizing the total energy consumption. Simulation results show that Pareto-optimal solutions can provide significant energy savings at the cost a slight increase in the probability of error
from its minimum achievable value.
Having detected the presence of the source, accurate source localization is another important task to be performed by a WSN. The state-of-the-art one-shot location es-timation scheme requires simultaneous transmission of all sensor data to the fusion center. We propose an iterative source localization algorithm where a small set of anchor sensors first detect the presence of the source and arrive at a coarse location estimate. Then a number of non-anchor sensors are selected in an iterative manner to refine the location estimate. The iterative localization scheme reduces the communi-cation requirements as compared to the one-shot locommuni-cation estimation while introducing some estimation latency. For sensor selection at each iteration, two metrics are proposed which are derived based on the mutual information (MI) and the posterior Cram´er-Rao lower bound (PCRLB) of the location estimate. In terms of computational complexity, the PCRLB-based sensor selection metric is more efficient as compared to the MI-based sensor selection metric, and under the assumption of perfect communication channels between sensors and the fusion center, both sensor selection schemes achieve the simi-lar estimation performance that is the mean squared error of the source location gets very close to the PCRLB of one-shot location estimator within a few iterations. The proposed iterative method is further extended to the case which considers fading on the channels between sensors and the fusion center. Simulation results are presented for the cases when partial or complete channel knowledge are available at the fusion center.
We finally consider a heterogenous sensing field and define a distributed parameter estimation problem where the quantization data rate of a sensor is determined as a function of its observation SNR. The inverse of the average Fisher information is then defined as a lower bound on the average PCRLB which is hard to compute. The inverse of the average Fisher information is minimized subject to the total bandwidth and bandwidth utilization constraints and we find the optimal transmission probability of each possible quantization rate. Under stringent bandwidth availability, the proposed scheme outperforms the scheme where the total bandwidth is equally distributed among sensors.
TELS˙IZ DUYARGA A ˇGLARI ICIN
RASTLANTISAL OLAYLARIN KAYNAK DUYARLI DA ˇGINIK TESB˙IT VE KEST˙IR˙IMI
Engin Ma¸sazade
Doktora Tezi, 2010
Tez Danı¸smanı: Do¸c. Dr. Mehmet Keskin¨oz Tez Ek Danı¸smanı: Prof. Dr. Pramod K. Varshney
Anahtar Kelimeler: Telsiz duyarga aˇglari, daˇgınık tesbit ve kestirim prob-lemleri, ¸cok ama¸clı eniyileme, duyarga se¸cimi, s¨on¨umlemeli kanallar, tel-siz haberle¸sme
Bu tezde, telsiz duyarga aˇglar icin da˘gınık tesbit ve kestirim problemleri kay-nak duyarlılıˇgˇi altında incelenmi¸stir. Duyargalar ufak, pille beslenen cihazlar oldugun-dan, kaynakların (enerji, bandgeni¸sliˇgi) tasarruflu kullanımı ¨onemlidir. Bu doˇgrultuda ula¸sılabilen en iyi ba¸sarımdan fazla ¨od¨un vermeden kaynaklardan onemli ¨ol¸cude tasar-ruf eden y¨ontemler sunulmaktadır. ˙Ilk olarak daˇgınık tesbit sorunu ele alınmı¸stır. Ama¸cların t¨umle¸stirme merkezi karar hata olasılıgının ve aˇgın toplam enerji sarfiyatının en aza indirgenmesi olduˇgu bir ¸cok ama¸clı en iyileme problemi tanımlanmı¸stır. Bu prob-lemden elde edilen sonu¸clar, ula¸sılabilecek en d¨u¸s¨uk hata olasılıˇgına yakın ama ¨onemli
¨ol¸c¨ude enerji tasarrufu saˇglayan karar e¸siklerinin de olduˇgunu g¨ostermi¸stir.
Olayın varlıˇgının tesbiti kadar olay yerinin hassas kestirimi t¨url¨u uygulamalar a¸cısından ¨onemlidir. T¨um duyarga verisini tek bir defada g¨ondermek yerine, tekrarlı bir kestirim y¨ontemi sunulmaktadır. Az sayıda duyarganın verisi kullanılarak olay yeri ¨once kabaca kestirilir. Y¨ontemin bir diˇger tekrarında, verisi istenecek duyargalar m¨u¸sterek bilgi veya sonsal Cram´er-Rao alt sınırı esaslı metrikler yardımıyla se¸cilmektedir. M¨u¸sterek bilgi veya sonsal Cram´er-Rao alt sınırı temelli duyarga se¸cim metrikleri kusursuz ile-tim kanalları varsayımı altında benzer kestirim ba¸sarımı g¨osterseler de, hesaplarımız sonsal Cram´er-Rao alt sınırı temelli duyarga se¸cim metriˇginin karma¸sıklıˇgının m¨u¸sterek bilgi temelli duyarga se¸cim metriˇgine g¨ore daha az olduˇgunu g¨ostermi¸stir. Benzetim sonu¸clarımız, tekrarlı olay yeri kestirimi y¨onteminin, ufak bir gecikme pahasına, kestirim i¸cin aˇgdaki haberle¸sme gereksinimini t¨um duyarga verisini istemeye g¨ore ¨onemli ¨ol¸c¨ude azalttıˇgını g¨ostermi¸stir. ¨Onerilen tekrarlı olay yeri kestirim y¨ontemi s¨on¨umlemeli kanal-lar i¸cin de genelle¸stirilmi¸stir. ¨Oncelikle, kestirimde temel ba¸sarım kıstası olan sonsal Cram´er-Rao alt sınırı tum duyarga verisi icin t¨uretilmi¸s, saydıˇgımız iki duyarga se¸cim metriˇgi t¨um ve kismi kanal bilgisi varsayımları altında yinelenmi¸stir.
Son olarak, ¨ol¸c¨um g¨ur¨ult¨us¨un¨un her bir duyarga i¸cin farklı olduˇgu ayrı¸sık du-rum incelenmi¸stir. Her bir duyarganın ol¸c¨um¨un¨u temsilde kullandıˇgı kuantalama hızı, ¨ol¸c¨um¨un¨un i¸saret-g¨ur¨ult¨u oranlarına baˇglı olarak belirlenmı¸stır. Verilen bandgeni¸sliˇgi altında, genel bir daˇgınık kestirim problemi incelenmi¸stir. Toplam bandgeni¸sliˇgini a¸smamak icin belirli bir kuantalama hızında temsil edilen ¨ol¸c¨um, t¨umle¸stirme merkezine yine belirli bir g¨onderim olasıˇgı ile iletilmektedir. Her bir kuantalama hızının, en uy-gun g¨onderim olasılıˇgını bulmak i¸cin toplam bandgeni¸sliˇgi ve band kullanımı kıstasları altında ortalama Fisher bilgisinin tersi en aza indirgenmi¸stir. Benzetim sonu¸cları, kısıtlı bandgeni¸sliˇginde, ¨onerdiˇgimiz y¨ontemin kestirim hatasını bandgeni¸sliˇgini duyargalar arasında e¸sit olarak b¨ol¨u¸st¨uren y¨onteme g¨ore olduk¸ca d¨u¸s¨urd¨uˇg¨un¨u g¨ostermektedir.
Acknowledgments
I would like to thank my advisor Prof. Mehmet Keskin¨oz for his invaluable guidance and support throughout my graduate study at Sabancı University. I am heartily thankful to my thesis co-advisor, Pramod K. Varshney, whose inspiration, encouragement, guidance and support enabled me not only to develop the understanding of the research topic but also to improve my personal skills. I would like to express the deepest appreciation to Prof. Ruixin Niu who always made time for every question I had, and I really enjoyed discussing my research with him. This dissertation would not have been possible without their guidance.
Also, I would like to thank Prof. G¨ull¨u Kızılta¸s S¸endur for being a defense com-mittee member and providing an excellent research direction on Multiobjective opti-mization methods. I would like to thank Prof. M¨ujdat C¸ etin also for being a committee member and I appreciate him for the insights he has shared. I would like to thank Dr. Ramesh Rajagopalan for introducing me the multiobjective optimization methods, Dr. Hao Chen and Prof. Chilukuri K. Mohan for their invaluable comments and sugges-tions.
I would like to thank all my friends from Sabancı and Syracuse Universities. I would like to thank my friends at Sensor Fusion Lab. Priyadip, Arun, Onur, ˙Ilker, Thakshila, Swarnendu, Satish, Ashok, Renbin and Long for their wonderful friendship and the research environment. Also, I would like to thank my friends from Commu-nications and Networking Lab., Kayhan, Ali, Mehmet and Yunus for being the best office mates. Last but not least, I would like to thank my dear friends Ertuˇgrul, Osman G¨okhan, Can, Sarper, Sel¸cuk, Erdem, Eray, Hadi, G¨okhan, ¨Ozlem, Aslı, H¨ulya, Hatice
and ¨Oznur for their invaluable friendship and support during the hard years of study. I wish you all the success, fortune and happiness in the world.
Finally, and most importantly, I would like to thank my parents Zehra and Rıfat Ma¸sazade and my sister Deniz. I would not be here without them and without their unconditional love, patience and support. I send my love to Furkan, Hilal, and Fatih whom I have missed watching them grow up.
This work is partially supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under the Research Abroad Support Scheme, TUBITAK research grant under 105E161, ARO Grant W911NF-09-1-0244, and AFOSR under grant FA-9550-06-1-0277.
TABLE OF CONTENTS
1 Introduction 1
1.1 Preliminaries . . . 6
1.1.1 Bayesian Detection Theory . . . 6
1.1.2 Estimation Theory . . . 8
1.1.3 Information Measures . . . 9
1.1.4 Monte Carlo Methods . . . 11
1.1.5 Multiobjective Optimization . . . 13
1.2 Research Motivation and Approach . . . 13
1.3 Major Contributions and Dissertation Organization . . . 17
1.4 Notes . . . 20
2 A Multi-objective Optimization Approach to Obtain Decision Thresh-olds for Distributed Detection 22 2.1 Problem Definition . . . 26
2.1.1 Wireless Sensor Network Model and Statement of the MOP . . 26
2.1.2 Parallel Decision Fusion . . . 29
2.1.3 Serial Decision Fusion . . . 33
2.2 Multiobjective Optimization . . . 38
2.2.1 Normal Boundary Intersection (NBI) . . . 38
2.2.2 Non-Dominating Sorting Genetic Algorithm - II (NSGA-II) . . . 42
2.2.3 Performance Metrics . . . 45
2.3.1 Simulation Settings . . . 46
2.3.2 Performance Comparison of NBI and NSGA-II . . . 47
2.3.3 Optimal Pareto Fronts . . . 48
2.3.4 The Performance of WSN . . . 50
2.3.5 Identical Decision Thresholds . . . 53
2.4 Discussion . . . 56
3 Energy Aware Iterative Source Localization 58 3.1 System Model . . . 62
3.1.1 Source detection using multi-bit sensor data . . . 64
3.1.2 Source location estimation using multi-bit sensor data . . . 65
3.2 Iterative Source Location Estimation Method . . . 66
3.2.1 Source Location Estimation Based on Monte Carlo Methods . . 69
3.2.2 Sensor Selection Methods . . . 70
3.3 Sensor Data Compression . . . 80
3.4 Simulation Results . . . 82
3.4.1 Source Detection Performance . . . 83
3.4.2 Computational Cost . . . 85
3.4.3 Iterative Location Estimation Performance . . . 86
3.4.4 The trade-off between estimation performance and communica-tion cost . . . 91
3.5 Discussion . . . 93
4 Channel Aware Iterative Source Localization 102 4.1 System Model . . . 103
4.1.1 WSN assumptions . . . 103
4.1.2 CRLB of the source location estimate . . . 106
4.2 Iterative Source Location Estimation under Channel Fading . . . 108
4.2.1 PCRLB of the source location estimate . . . 108
4.2.2 Source localization using a Sequential Monte-Carlo method . . . 109 4.2.3 Mutual Information Based Sensor Selection under Channel Fading110
4.2.4 PCRLB Based Sensor Selection under Channel Fading . . . 113
4.3 Simulation Results . . . 116
4.3.1 Performance of the one-shot location estimator . . . 117
4.3.2 Performance of the iterative location estimation . . . 118
4.4 Discussion . . . 120
5 A Probabilistic Rate Transmission Scheme for Distributed Estimation126 5.1 Problem Formulation . . . 128
5.1.1 Data rate decision of each sensor . . . 129
5.1.2 Determination of the quantization thresholds . . . 132
5.2 Probabilistic Rate Transmission . . . 133
5.2.1 The average FI of the estimate . . . 133
5.2.2 Total Rate Constraint . . . 136
5.2.3 Bandwidth utilization constraint . . . 136
5.3 Parameter Estimation based on Received Sensor Data . . . 139
5.3.1 The Likelihood under probabilistic rate transmission . . . 140
5.3.2 The Likelihood under equal rate transmission . . . 140
5.4 Simulation Results . . . 141
5.5 Discussion . . . 142
6 Concluding Remarks and Suggestions for Future Work 144 A Proof of Lemmas 147 A.1 Proof of Lemma 4.1 . . . 147
LIST OF TABLES
2.1 Generational Distance between NBI and NSGA-II, Spread Metric and
Mean Execution Times (E.T.) for NSGA-II and NBI. . . 47
2.2 Domination Metric between NBI and NSGA-II. . . 48
3.1 The sensor layouts to evaluate the detection performance. . . 83
3.2 Mean CPU times of MI and PCRLB . . . 86
3.3 Final MSE at the end of the 9th iteration. A = 1, PCRLB based sensor selection. . . 90
3.4 A = 1, Average number of bits used to represent the M = 3, M = 4, M = 5 and M = 6 bit sensor data. . . . 90
4.1 Estimation Improvement by increasing M, ²b = 1 . . . 117
4.2 Estimation Improvement by increasing M, ²b = 5 . . . 118
5.1 Optimal transmission probabilities of each possible transmission rate Rk = j bits. (Conservative case: ² = 0.01, δ = 0.5) . . . . 142
5.2 Optimal transmission probabilities of each possible transmission rate Rk = j bits. (Liberal case: ² = 0.1, δ = 0.7) . . . . 142
LIST OF FIGURES
1.1 An example wireless sensor network . . . 3 1.2 System model for sensor and resource management based on feedback
from recursive estimator. . . 6 2.1 Wireless Sensor Network Model with Parallel Decision Fusion . . . 27 2.2 Wireless Sensor Network Model with Serial Decision Fusion . . . 34 2.3 The Pareto optimal front found by minimizing the weighted sum of the
objective functions, N = 5 . . . . 39 2.4 The point P is the solution of the single-objective constrained NBI
sub-problem outlined with the dashed line v . . . . 41 2.5 Contour lines of the objective functions with N = 2 Sensors, parallel
configuration (a) Probability of Error (b) Energy Consumption . . . 42 2.6 Contour lines of the objective functions with N = 2 Sensors, serial
con-figuration (a) Probability of Error (b) Energy Consumption . . . 43 2.7 Pareto Optimal Solutions generated via NBI and NSGA-II methods for
parallel fusion and non-identical decision thresholds at each sensor . . . 49 2.8 Pareto Optimal Solutions obtained by NBI and NSGA-II methods for
serial fusion and non-identical decision thresholds at each sensor. . . 50 2.9 Parallel Decision Fusion - Local sensor error probability as a function of
its mean distance to the event location . . . 52 2.10 Parallel Decision Fusion - Local sensor energy consumption as a function
2.11 Serial Decision Fusion - Global error probability as a function of the hop count on the routing path . . . 54 2.12 Serial Decision Fusion - Global energy consumption as a function of the
hop count on the routing path . . . 55 2.13 Parallel Decision Fusion, Pareto Optimal Solutions for identical and
non-identical sensor thresholds . . . 56 2.14 Serial Decision Fusion, Pareto Optimal Solutions for identical and
non-identical sensor thresholds . . . 57 3.1 The signal intensity contours of a source located in a sensor field. . . . 62 3.2 Wireless Sensor Network Model. Black Points: Sensor Locations; Blue
Squares: Anchor Sensors used for initial iteration; Green Circles: Acti-vated Sensors after 10 iterations for the example considered in Section V; Red Star: Source. A = 1 sensor is activated / iteration . . . . 67 3.3 The flow chart of the algorithm. The dashed blocks represent the
state-of-the-art Mutual Information based sensor selection method. The entire set of solid blocks represent the PCRLB based algorithm. . . 68 3.4 Conditional Entropy of non-anchor sensors in the field given the
multi-bit decisions of the anchor sensors at the beginning of the first iteration. (a) M = 5 bit, (b) M = 6 bit . . . . 81 3.5 Grid Sensor Layout specified by inter-sensor distance (ISD) and distance
of the nearest sensor to the mean source location (D-NS-MSL). (Black points: Sensors, Red Star: Source) . . . 84 3.6 Detection performance of anchor sensor layouts . . . 85 3.7 Mean computation times of objective functions . . . 86 3.8 Γ(1, 1) and Γ(2, 2) at the first iteration as a function of δ (√G = 100,
M = 5). . . . 88 3.9 The MSE performance of the PCRLB based sensor selection.
Compari-son of Numerical computation and Gaussian approximation for the FIM of the prior. (a) M = 3, (b) M = 4, (c) M = 5 . . . 96
3.10 MSE at each iteration sensor selection is based on MI, PCRLB and nearest sensor to the estimated source location. (a) M = 3, (b) M = 4, (c) M = 5, (d) M = 6 bits quantization . . . . 98 3.11 (a) MSE performance of MI and PCRLB based sensor selection schemes.
N = 361, K = 16, M = 5, A = 2; (b) M = 5 bit quantization of each sensor measurement, MSE performance of source localization with PCRLB based sensor selection and data compression. A = 1, A = 2 and A = 3 sensor activations / iteration. . . . 99 3.12 Stopping metric vs. the number of sensors to be selected. The black line
with triangle markers indicates the accuracy threshold. (i = 1, ² = 5) . 100 3.13 (a) Average number of iterations until the termination of the algorithm.
(b) Average number of bits transmitted to the fusion center until the termination of the algorithm. (M = 5, M = 6, ² = 5, 100 different trials.)101 4.1 Trace of the MSE matrix of N sensor data, (a) ²b = 1, (b) ²b = 5 (M = 3)121
4.2 MSE performance of the iterative scheme using MI-based sensor selection (a) ec = 1 (b) ec = 5, (The dashed line is the trace of the MSE matrix
of N = 49 sensor data.) . . . . 122 4.3 Mean channel gain |hk|; mean distance between the source location and
the selected sensor |dk| at each iteration (MI-based sensor selection). . . 123
4.4 Evaluation of the prior FIM as a function of δ, (M = 3, ROI = 20 × 20) 123 4.5 MSE performance of the iterative scheme using MI and PCRLB-based
sensor selection (a) ²b = 1 (b) ²b = 5, (The dashed line is the trace of the
MSE matrix of N = 49 sensor data.) . . . . 124 4.6 (a) Mean channel gain and (b) Mean distance to the source location
of the sensor selected at each iteration (MI and PCRLB based sensor selection). . . 125 5.1 Fisher information of a single sensor at various SNRk. Dashed lines
represent the FI for the case where a sensor transmits analog data and M = 6. . . . 131
5.2 Optimal data rate which maximizes the FI per bit vs. τk. M = 6, µθ = 0
and σ2
θ = 10. . . 132
5.3 MSE of probabilistic vs equal rate transmission schemes, µθ = 0 and
σ2
Chapter 1
Introduction
A wireless sensor network (WSN) consists of a large number of spatially distributed sensors that have signal processing abilities. Sensors have finite battery lifetime and thus limited computing and communication capabilities. When properly programmed and networked, sensors in a WSN cooperate to perform different tasks that are use-ful in a wide range of applications such as battlefield surveillance, environment and health monitoring, and disaster relief operations. Therefore, WSNs have recently been considered as an attractive low-cost technology for a wide range of surveillance and monitoring applications [1].
A WSN may be employed to monitor the occurrence of random events in a variety of applications. A random event may occur at an unknown time, at an unknown location in a region of interest (ROI) or one of its attributes (such as energy or frequency) can be random and may vary in time. However, we may have a statistical model for the event or may be able to learn it. As an example, many random arrival or counting processes follow the Poisson distribution. The time interval of interest can be of any length like seconds, minutes, or years. Examples of temporal Poisson counting processes might include the number of illegal border crossing per day or the number of earthquakes per year. Examples of spatial Poisson counting processes might include the number of people per square mile/kilometer or the number of border crossing attempts per mile/kilometer [2]. Failures represent another class of interesting events. Component failures may lead to system failures if they are not detected and corrected. For example,
a truss on a bridge may buckle or a solenoid in a printer may burn out. The Weibull distribution can be used to model the time to failure of a component, measured from some specified time until the component fails [2]. If the task of the WSN is to monitor room temperature, slight deviations from the desired temperature of the room can be modeled using a Normal (Gaussian) distribution. In a battlefield scenario, the location of a source transmitting energy by a friend or foe unit can be assumed to be uniformly distributed over the entire region of interest (ROI).
In this dissertation, we focus on distributed detection and estimation of random events in WSNs. In distributed detection, multiple sensors work collaboratively to distinguish between two hypotheses such as the absence or presence of an event. In dis-tributed estimation, an underlying event or a specific attribute of the event is estimated based on the sensor observations. Since sensors are tiny battery powered devices with limited signal processing capabilities, prolonging the lifetime of a WSN is important for both commercial and tactical applications. With non-rechargeable batteries, this requirement places stringent energy constraints on the design of all WSN operations. Energy limitation is one of the major differences between a WSN and other wireless networks such as wireless local area networks. Also, WSNs are often self-configured networks with little or no pre-established infrastructure as well as a topology that can change dynamically. Moreover, there may be channel impairments such as fading and path loss in the network environment that can considerably degrade the quality of wire-less links among sensors. Such challenges should be taken into account while designing the communication and local signal processing algorithms for the WSN. For instance, to maximize battery lifetime and reduce communication bandwidth, it is essential for each sensor to locally compress its observed data so that only low rate inter-sensor or sensor to fusion center communication is required.
The design of distributed detection and estimation algorithms depends on the underlying WSN topology. In the literature, several popular WSN deployments charac-terized by the presence or absence of a fusion center have been considered. In a parallel fusion topology [3], [4], [5], [6] there is no inter-sensor communication, that is the com-munication is only between sensors and the fusion center. The fusion center collects
Figure 1.1: An example wireless sensor network
locally processed data and produces a final inference. In an ad hoc WSN [7], [8], there is no fusion center. The network itself is responsible for processing the collected infor-mation, and sensors communicate with each other through the shared wireless medium and arrive at a consensus. Furthermore, hybrid schemes are also possible in which the WSN is partitioned into clusters with a hierarchical structure [9], [10]. Each cluster has a local fusion center generating intermediate estimates, which are combined to obtain a final result at the global fusion center.
In this dissertation, we consider the case where sensors communicate directly with the fusion center. In Fig. 1.1, we show an example WSN with N distributed sensors where the sensor measurements or their compressed versions are directly transmitted to a central fusion center. Let sk represent a sensor where k ∈ {1, 2, ..., N }. A sensor
receives a noisy measurement zk from the event of interest θ which has the form,
zk = g(θ) + nk (1.1)
received signal at each sensor is subject to path loss, the observation function g(.) should be defined using a suitable path loss model which we define later in the dissertation. We assume that the observation noise nk is generated from the Gaussian distribution
and is independent across sensors.
Since the sensors suffer from severe energy, computation and storage limitations, the transmission of raw measurements to the fusion center is not desirable as it incurs excessive energy and bandwidth consumption. In distributed detection or estimation, the sensors first locally process their measurements and quantized versions of the deci-sion statistics are sent to the fudeci-sion center for making the final inference. The quantized measurement of each sensor Dk is obtained from the raw measurement zk according to,
Dk= 0 −∞ < zk < ηk,1 1 ηk,1 < zk< ηk,2 ... L − 2 ηk,L−2 < zk < ηk,L−1 L − 1 ηk,L−1 < zk < ∞ (1.2)
where ηk = [ηk,0, ηk,1, . . . , ηk,L]T is the set of quantization thresholds for sk with ηk,0 =
−∞ and ηk,L = ∞.
Let rk be the received data of each sensor at the fusion center. The quantized
data of sensors are either assumed to be sent to the fusion center over perfect communi-cation links (rk = Dk) [11] or over imperfect channels (rk = h(Dk)) where the channel
impairments h(.) are modeled by suitable channel fading and noise models [4], [12]. The task of the fusion center is to make an inference about the event of interest. If the fusion center is responsible to detect either the absence or the presence of the event, the problem is called the detection problem. Let D0 be the binary decision of
the fusion center, which is defined as follows,
D0 =
0 Fusion center decides on H0
1 Fusion center decides on H1
where Hypothesis 0 (H0) and Hypothesis 1 (H1) denote the absence and the presence
of the event.
Moreover, if the fusion center is responsible for estimating an attribute of the event of interest (θ), the problem is called an estimation problem. Let T be an estimator which is a function of the received sensor data R = [r1 r2 ... rN], then the estimate of
the parameter is represented as ˆθ and obtained as,
ˆ
θ = T (R) (1.4)
If the event of interest θ is modeled as an unknown constant, under certain regularity conditions, the optimal estimator is the maximum-likelihood (ML) estimator [13]. If the unknown parameter θ has a prior distribution, then the Bayesian estimator minimizes the Bayes risk [13]. We leave the details of the estimators for later in the dissertation. Since a wireless sensor network consists of densely deployed tiny, battery-powered sensors, they have limited on-board energy. Therefore, if a sensor remains continuously active, its energy will be depleted quickly. In order to prolong the network lifetime, sensors should alternate between being active and idle. Note that dense deployment of sensors brings redundancy in coverage. Therefore, selecting a subset of sensors, can still provide information with the desired quality. The sensor management policies define the selection of active sensors to meet the application requirements while minimizing the use of resources [14], [15], [16], [17], [18], [19]. In other words, the problem of adaptive sensor management and resource allocation in sensor networks is to determine the optimal way to manage system resources and task a group of sensors to collect measurements for statistical inference. As shown in Fig. 1.2, the distributed sensors send their observations through band-limited channels to a fusion center, where the data are fused by an estimator to update the object state estimate. The updated state estimate based on the past data is used to guide the sensor management and resource allocation procedure adaptively. Sensor management is carried out in a way such that at the next time step, the best estimation accuracy is achieved under pre-specified resource utilization constraints.
Figure 1.2: System model for sensor and resource management based on feedback from recursive estimator.
1.1 Preliminaries
In this section, we briefly summarize the necessary background for several topics that will be considered in the dissertation. Since we deal with detection and estimation in wireless sensor networks, we first present the fundamentals of Bayesian detection and estimation theories. Then we review information measures such as entropy and mutual information. We then discuss Monte-Carlo based methods and finally present the problem formulation of Multi-objective optimization.
1.1.1 Bayesian Detection Theory
Let H0 and H1 denote the two hypotheses for the binary hypothesis testing problem.
Let the observation be denoted as z so that the conditional densities under the two hypotheses are p(z|H0) and p(z|H1) respectively. The observations are generated with
these conditional densities which are assumed known. The a priori probabilities of the two hypotheses are denoted by P (H0) and P (H1) respectively. In the binary hypothesis
testing problem, four possible actions can occur. Let Ci,j, i ∈ {0, 1}, j ∈ {0, 1} represent
the cost of declaring Hi true when Hj is present. The Bayes risk function is given by,
R =P1i=0P1j=0Ci,jP (Hj)P (Decide Hi|Hj is present) (1.5)
=P1i=0P1j=0Ci,jP (Hj)
R
where Zi is the decision region corresponding to hypothesis Hi which is declared true
for any observation falling in the region Zi. Let Z be the entire observation space so
that Z = Z0
S
Z1 and Z0
T
Z1 = ∅.
Collecting the terms in (1.5) yields,
R = P (H0)C1,0+ P (H1)C1,1+ (1.6)
Z
Z0
{[P (H1)(C0,1− C1,1)p(z|H1)] − [P (H0)(C1,0− C0,0)p(z|H0)]}
The risk is minimized by assigning those points of Z to Z0that make the integrand
of (1.6) negative. Then, the minimization results in the likelihood ratio test (LRT) p(z|H1) p(z|H0) ≷H1 H0 P (H0)(C1,0− C0,0) P (H1)(C0,1− C1,1) (1.7)
The quantity on the left hand side is known as the likelihood ratio and the quantity on the right hand side is the threshold. In this dissertation, we consider Bayes criterion for the design of decision rules. The minimax criterion [3] is a good alternative when the knowledge of a priori probabilities of each hypothesis is not available. Moreover, in many practical applications not only the a priori probabilities but also the cost assignments are difficult to make. For such cases, Neyman-Pearson test is employed that constrains the probability of false alarm (PF) while maximizing the probability of
detection (PD). Then, PF and PD are defined as,
PF = P (Decide H1|H0 is present) = Z Z1 p(z|H0)dz PD = P (Decide H1|H1 is present) = Z Z1 p(z|H1)dz
A more detailed treatment of detection theory can be found in a wide variety of books such as [3] and [20].
1.1.2 Estimation Theory
Assume that a scalar parameter θ to be estimated using the vector of measurements z. Let p(z; θ) be the probability distribution function (pdf) of z given θ and it is assumed that p(z; θ) satisfies the regularity condition [21],
E · −∂ ln p(z; θ) ∂θ ¸ = 0 for all θ (1.8)
where the expectation is taken with respect to p(z; θ). Then the variance of any unbiased estimator ˆθ must satisfy,
var(ˆθ) ≥ 1 E
h
−∂2ln p(z;θ)∂θ2
i (1.9)
where the expectation is taken with respect to p(z; θ) and the derivative is evaluated at the true value of θ.
Let θ = [θ1, . . . , θN] be a vector parameter to be estimated. Then, the variance
of each element θi is found by the inverse of the Fisher information matrix defined as,
var(ˆθi) ≥
£
F−1(θ)¤i,i (1.10)
where i, jth element of F (θ) is defined as,
[F (θ)]i,j = E ·
−∂2p(z; θ) ∂θi∂θj
¸
Let the vector of unknown parameters θ be a random vector with pdf p(θ). Then, the minimum mean squared error (MSE) estimator is the conditional mean estimator defined as,
ˆ θMSE =
Z
θp(θ|z)dθ (1.11)
In Bayesian estimation, the performance of any estimator ˆθ(z) can be bounded by the Posterior Cram´er - Rao Lower Bound, under suitable regularity conditions [13]. The variance of each element θi is found by the inverse of the Fisher information matrix J
whose i, jth element is found according to, Ji,j = E · −∂2p(z; θ) ∂θi∂θj ¸ (1.12)
where the expectation is taken with respect to p(z, θ) which is the joint entropy between z and θ. The PCRLB states that,
var(ˆθi) ≥
£
J−1(θ)¤i,i (1.13)
with equality if and only if the a posteriori density p(θ|z) is a multivariate Gaussian density [13].
1.1.3 Information Measures
In this section, we give basic definitions and properties of several information mea-sures such as entropy, conditional entropy and mutual information that we use in the dissertation. More details can be found in [22] and [23].
Entropy
The entropy or uncertainty of the discrete random variable X is,
H(X) = X
a∈sup(PX)
−PX(a) log2PX(a) (1.14)
where the support of a random variable X is the set,
sup(PX) = {a : a ∈ X, PX(a) > 0} (1.15)
Alternatively we can write,
Conditional Entropy
The conditional entropy of X given the event Y = b with probability P (Y = b) > 0 is,
H(X|Y = b) = X
a∈sup(PX|Y(.|b))
−PX|Y(a|b) log2PX|Y(a|b)
= E[− log2PX|Y(X|Y )|Y = b] (1.17)
The conditional entropy of X given Y is the average of the values (1.17), that is,
H(X|Y ) = X
b∈sup(PY)
PY(b)H(X|Y = b)
= X
(a,b)∈sup(PX,Y)
−PX,Y(a, b) log2PX,Y(a, b)
= E[− log2PX|Y(X|Y )] (1.18)
Mutual Information
The mutual information I(X, Y ) between two random variables X and Y with respec-tive discrete and finite alphabets is defined as,
I(X, Y ) = H(X) − H(X|Y ) (1.19)
The name “mutual” describes the symmetry in the arguments of I(X, Y ), i.e,
I(X, Y ) = H(Y ) − H(Y |X) (1.20)
Note that,
I(X, Y ) ≥ 0 H(X|Y ) ≤ H(X) H(X, Y ) ≤ H(X) + H(Y )
with equality if and only if X and Y are statistically independent. The second inequality above means that conditioning reduces entropy.
1.1.4 Monte Carlo Methods
Suppose, we want to compute the following integral for the test function ϕ.
I = Z
Z
ϕ(z)dz (1.21)
where Z represents the integration range of ϕ. To compute I numerically, Monte Carlo methods introduces a new pdf π(z) as
I ≈ IA
IA=
Z
Z
ϕ(z)π(z)dz (1.22)
where IA is the numerical approximation of I. The Monte-Carlo method approximates
π by the following point-mass measure.
ˆ π(z) = 1 N N X i=1 δ(z − zi) (1.23)
where N is the number of samples. Using (1.23) in (1.22) yields,
IA = Z Z ϕ(z)ˆπ(z)dz = 1 N N X i=1 ϕ(zi) (1.24)
Suppose we are interested in sampling from π(z) and assume that we are able to sample from another pdf q(z). The importance sampling procedure allows us to sample from π using q(z). Assuming that π(z) > 0 and q(z) > 0, the following identity trivially holds,
where w(.) is the importance weight given by,
w(z) = π(z)
q(z) (1.26)
This suggests that if N samples {zi} from q(.) are available, then an approximation of
this distribution is given by,
ˆ q(z) = 1 N N X i=1 δ(z − zi) (1.27)
Plugging this approximation in (1.25), we obtain
ˆ π(z) = 1 N N X i=1 w(zi)δ(z − zi) (1.28)
Let wi be the normalized importance weights as,
wi = Pw(zi) N
j=1w(zj)
(1.29)
Using importance sampling, (1.24) is computed as follows,
IA= N
X
i=1
wiϕ(zi) (1.30)
In Chapter 3, based on the available data z, we are interested in approximating the posterior distribution
p(θ|z) ∝ p(z|θ)p(θ)
Note that the initial probability of each sample p(θi) = 1/N. The weights are then selected as the likelihood of the received samples as
w(zi) = p(z|θi)
1.1.5 Multiobjective Optimization
In this section, we briefly define the problem formulation of multiobjective optimization. The mathematical description of multiobjective optimization [25], [26], [27], [28], [29], [30], [31] can be given as follows:
min
χ∈C [f1(χ) f2(χ) ...fn(χ)]
T (1.31)
where χ is a candidate solution to the multiobjective optimization problem (MOP). The number of objectives n ≥ 2 and the feasible set C,
C : {χ : h(χ) = 0, g(χ) ≤ 0, a ≤ χ ≤ b} (1.32) is subject to the equality and inequality constraints denoted as h(χ) and g(χ) respec-tively, and explicit variable bounds [a, b]. In a minimization problem, a solution χ1
dominates another solution χ2 (χ1 À χ2) if and only if
fu(χ1) ≤ fu(χ2) ∀u ∈ {1, 2, .., n} (1.33)
fv(χ1) < fv(χ2) ∃v ∈ {1, 2, .., n}
and a solution χ∗ is the Pareto optimal solution for the MOP if and only if there is
no χ ∈ C that dominates χ∗. Pareto optimal points are also known as non-dominated
points. A well known technique for solving MOPs is to minimize a weighted sum of the objective functions. Later in the dissertation we utilize several different methods to obtain the Pareto-optimal front.
1.2 Research Motivation and Approach
The distributed detection problem for WSNs has been studied extensively. If the fusion center receives the raw measurements of sensors, the problem is reduced to the classic hypothesis testing problem [32]. For quantized sensor data, in the temporal asymptotic regime, information theoretic frameworks have been developed [33], [34], [35] to find
the optimal decision rules based on the error exponent. For the case where the number of sensors goes to infinity, it is shown in [36] and [37] that an identical decision rule for all the sensors is asymptotically optimal.
For practical systems with limited number of sensors, the distributed detection problem can be decomposed into two inter-related problems. The first problem is to find the optimal decision rule at the fusion center. This is a relatively simple problem since the optimal fusion rule reduces to a likelihood ratio test (LRT) for binary and multi-bit sensor decisions [3]. The second problem is to obtain the decision rules at the sensors which is more complicated. Under the conditional independence assumption, the optimal decision rule at each sensor is expressed as an LRT [3]. Since the decision rules at distributed sensors and the fusion center are dependent on each other, person-by-person optimization (PBPO) is often used to obtain the optimal decision thresholds of sensors [38]. Many papers in the literature, assume ideal channels between the sensors and the fusion center. In [39], [4], [6], [40], non-ideal channels have been assumed between the sensors and the fusion center in the distributed detection context. Without the conditional independence assumption, the distributed detection problem becomes very hard [41], [42], [43].
In this dissertation, we first study the event detection problem for sensor net-works under the isotropic signal emission model [44], [45] where the source location is assumed to be uniformly distributed in a given ROI. Given the source location and the assumption of independent identical noise distribution at each sensor, the optimal decision rules at the sensors and the fusion center are LRTs. When the source location is random and available only in terms of its probability distribution, the conditional independence assumption of sensor measurements and the optimality of LRT are no longer valid [45]. We assume that each sensor arrives at a binary decision about the event by comparing locally computed decision statistic with its decision threshold. The binary decision is then transmitted to the fusion center only if the presence of the event is decided [46], [47]. Therefore, the decision rules used at the sensors not only determine the decision error probability achieved by the WSN but also the total energy consumption of the WSN. In order to find the sensor decision thresholds, we
formu-late a multi-objective optimization problem (MOP) with two objectives, minimizing the probability of error at the fusion center and minimizing the total network energy consumption. Using a multi-objective optimization approach, we seek those solutions which provide significant energy savings as compared to the minimum error solution at potentially the cost of a slight increase in the best achievable probability of error of the network.
After the presence of the source emitting energy is detected by the WSN, an important task that needs to be performed is source localization, which is important for an accurate tracking of the target and higher level motion analysis. Under the isotropic signal emission model, the event (or source) location can be determined based on the energy readings of sensors [48], [44]. In [48] and [44], maximum likelihood (ML) based source localization approaches have been proposed by using analog and multi-bit (M-bit) sensor measurements respectively at the fusion center. Furthermore, in [49], the authors propose a joint detection and source localization scheme using the data received from all the sensors in the network. We call the source localization scheme which requires simultaneous data transmission from all the sensors to the fusion center as one-shot location estimation. One-one-shot location estimation introduces several challenges. First of all, the sensors that are far from the source location are not likely to carry useful information but they still consume energy to transmit information. Secondly, each sensor requires an independent channel for simultaneous data transmission to the fusion center. This assumption imposes a limitation on the number of sensors that the system can support in practice. In our approach, we assume that the source location is random and characterized by a multivariate Gaussian distribution whose covariance matrix is large so as to cover the entire ROI. In our model, rather than transmitting multi-bit data from all the sensors in the network to the fusion center, we first employ measurements from relatively few anchor sensors to detect the presence of the source and obtain a coarse location estimate. The non-anchor sensors do not transmit their measurements in the initial phase. Then, a few non-anchor sensors are activated at each step of an iterative procedure. Since only the most informative sensors about the source location are selected, the iterative algorithm is expected to provide significant
energy savings as compared to one-shot location estimation at the cost of some latency. Since source location is a random parameter which has a certain prior pdf, we consider posterior Cramer Rao lower bound (PCRLB) as the estimation benchmark for the mean squared error (MSE).
The lossless communication assumption between sensors and the fusion center is often not valid in practice. Since WSNs are resource constrained in terms of bandwidth and energy, increasing the transmission power of sensors or employing powerful error correction codes to ensure lossless communication may not always be feasible. Also, in a hostile environment, the power of the transmitted signal should be kept low to decrease the probability of interception or detection. Therefore, the iterative source localization method also helps in dealing with the channel impairments.
So far, we have assumed that the WSN is homogenous, i.e., the observation noise of each sensor is independent and identically distributed and all the sensors send the same amount of information to the fusion center. Next, we investigate a heterogonous WSN where the observation noise of each sensor is independent but not identically dis-tributed [50] and depending on the quality of sensor observations, each sensor transmits different amount of data to the fusion center [51]. We consider a distributed random pa-rameter estimation problem under a total bandwidth constraint. In the literature, the total rate-constrained distributed estimation problem has been investigated extensively (see [52] and references therein). Under rate constraints, the source coding problem has been studied by deriving the information theoretic achievable rate regions in [5]. If no prior is assumed for the estimation parameter, then the dynamic range of the parameter is assumed to be bounded within a certain interval. For such cases linear decentral-ized estimation schemes have been proposed for homogenous environments [53] and for heterogenous environments [51], [50]. Moreover, for 1-bit sensor data, [54] investigates the performance limit of distributed estimation systems where the dynamic range of the estimation parameter is assumed to be known. Also in [55], the authors assume that the sensor observations are bounded and they propose nonparametric distributed estimators based on the knowledge of the first N moments of sensor noises. Different from the papers discussed so far, we assume that the parameter to be estimated follows
a certain prior probability distribution, which requires a Bayesian estimator to be em-ployed at the fusion center and PCRLB is computed as a benchmark for estimation. In a heterogonous network, the complexity to compute PCRLB is high, which motivates us to find another lower bound on the MSE. Since we assume that the total bandwidth is limited, so as not to exceed the total bandwidth, each sensor sending data at a specific data rate employs a certain transmission probability to send data to the fusion center. Using this approach of only requesting data from the sensors with more informative observations, we show that an estimation performance close to having all sensor data can be obtained. Similar probabilistic approaches for resource-constrained distributed estimation have been recently introduced in [56], [57]. In such approaches, each sensor measurement is transmitted to the fusion center with a certain probability so the total cost of information transmission from sensors to the fusion center does not exceed the available capacity. In [56], the authors have employed a channel-aware transmission control where the transmission probability of each sensor is chosen according to the quality of its local observation and transmission channels. In [57], the optimal trans-mission rates have been obtained by minimizing the posterior Cramer-Rao lower bound (PCRLB) under a total energy constraint. We follow a similar probabilistic scheme, and assume that each sensor transmits its data with a certain transmission probability per each quantization data rate. Given the number of sensors in the network and total available bandwidth, the transmission probabilities of each quantization rate minimizes the inverse of the Fisher information.
1.3 Major Contributions and Dissertation Organization
In this dissertation, resource aware distributed detection and estimation of random events in WSNs are investigated. We develop novel distributed detection and estima-tion schemes which can significantly save resources in terms of energy, communicaestima-tion and bandwidth while achieving a similar performance as the state-of-the-art detec-tion/estimation methods at the cost of potentially slight increase in the probability of decision error, estimation latency and outage probability.
source location is assumed to be uniformly distributed in a ROI. We formulate a multi-objective optimization problem (MOP) with two conflicting multi-objectives, minimizing the probability of error at the fusion center and minimizing the total network energy con-sumption. The decision thresholds at the sensors are selected as the optimization parameters of the MOP. We solve the MOP and generate the Pareto optimal solu-tions between these two conflicting objectives through Normal Boundary Intersection (NBI) [25] and Non Dominating Sorting Genetic Algorithm II (NSGA - II) [28]. Simu-lation results show that, instead of minimizing the global probability of error only, the proposed MOP approach provides a number of alternative solutions which are able to provide significant energy savings as compared to the minimum error solution at the cost of a slight increase in the minimum achievable probability of error of the network. In Chapter 3, we study the source localization problem for a homogenous WSN where the observation noise is independent and identically distributed for each sensor and all the sensors send the same amount of information to the fusion center. The source location is random and modeled using a multivariate Gaussian distribution whose covariance matrix is large so as to cover the entire ROI. We present an iterative source localization method where rather than transmitting complete sensor data to the fusion center from all the sensors, the anchor sensors first detect the source and obtain a coarse source location estimate. Then, we develop and compare two different sensor selection schemes for static source localization. The first scheme iteratively activates the non anchor-sensors which maximize the mutual information between source location and the quantized sensor measurements. In the second sensor selection scheme, a number of non-anchor sensors are activated whose quantized data minimize the PCRLB at each iteration. Further, using the posterior probability distribution function of the source location, we compress the quantized data of each activated sensor using distributed data compression techniques. Simulation results show that the MI and PCRLB based sensor selection schemes, within a few iterations achieve similar estimation performance and get close to the PCRLB for the case when all the sensor data are used. The PCRLB-based sensor selection is better in terms of computational complexity when the number of non-anchor sensors selected at each iteration is greater than one. By selecting only
the most informative sensors about the source location, the iterative approach provides large energy savings as compared to one shot location estimation while introducing some latency.
In Chapter 4, we extend the iterative source localization method for the case where the channels between sensors and the fusion center are subject to Rayleigh fading. Considering phase coherent reception and using the channel gain statistics, we first derive the likelihood of the M-bit symbols of a sensor received over a fading channel. Simulation results show that source location estimation using the channel gain statistics yield performance that is quite close to the case where each sensor’s channel gain is known exactly. We then extend the mutual information and PCRLB based sensor selection metrics that include channel fading. When the channel signal-to-noise ratio (SNR) is relatively high between sensors and the fusion center, the mean squared error of the iterative algorithm, in a few iterations gets close to the mean squared error when all N sensor data is available at the fusion center. On the other hand, if the channel SNR is low, then each selected sensor becomes less informative about the source location and the iterative sensor selection needs several iterations to reach the mean squared error of the case where data from all the N sensors is available.
In Chapter 5, we study a distributed parameter estimation problem for a hetero-gonous WSN where the observation noises of the sensors are Gaussian with non-identical statistics. The fusion center is unaware of the quality of the sensor observations and each sensor quantizes its measurement to the rate which improves its Fisher information per bit the most. For a heterogonous WSN, the complexity to compute the average PCRLB is high. To reduce the complexity associated with the PCRLB, we first show that the inverse of average Fisher information is a lower bound on the average PCRLB. From the previous chapter, we observe that the quantized sensor measurements become more informative as the wireless channel impairments are suppressed by increasing the energy per bit. In this chapter, we neglect the channel impairments for multi-bit sensor data, and assume the wireless channels between sensors and the fusion center are error free which can be provided by orthogonal channels with sufficient transmit power or powerful forward error correction. At the same time, we consider that the channels
between sensors and the fusion center can reliably transmit up to B bits information. So, not to exceed the total bandwidth (B), the observation of each sensor quantized with the rate computed as above is sent to the fusion center with a certain transmission probability. To find the optimal transmission probabilities of each possible data rate of a sensor, we formulate a constrained optimization problem by minimizing the inverse of the average Fisher information while taking the total bandwidth and network uti-lization constraints into account. Under stringent constraint on available bandwidth, simulation results show that the proposed probabilistic scheme, overcomes the scheme where the total bandwidth is equally distributed among all sensors in the network. In-stead of all sensors transmitting at high data rates which requires a large bandwidth, the proposed probabilistic bit transmission scheme obtains a similar MSE by requesting data at high rates only from the sensors with high SNR.
In Chapter 6, we summarize the main results of the dissertation and present suggestions for some future work.
1.4 Notes
We make use of the standard notational conventions. Vectors and matrices are written in boldface and all vectors are column vectors. For a matrix A, AT indicates the
trans-pose operation. The notation x ∼ N (µ, Σ) means that vector x is Gaussian distributed with mean vector µ and covariance matrix Σ. Also, throughout the dissertation, we denote the probability mass function of discrete variables by P (.) and the probability density function of continuous variables by p(.) or p(., .) depending on the number of random variables.
Portions of the material in this dissertation have been presented at the 2008 IEEE Asilomar Conference on Signals, Systems, and Computers [58], the 2009 International Workshop on Computational Advances in Multi-Sensor Adaptive Processing [59], the 2010 Conference on Information Sciences and Systems [60] and accepted for presentation at the 2010 International Conference on Information Fusion [61]. Additionally, portions of the material have appeared in or have accepted to appear in the IEEE Transactions on Systems, Man, and Cybernetics, Part B [62] and IEEE Transactions on Signal
Chapter 2
A Multi-objective Optimization Approach
to Obtain Decision Thresholds for
Distributed Detection
In this chapter, we study the detection problem where the objective of the WSN is to distinguish between two hypotheses, such as the absence (Hypothesis 0) or presence (Hypothesis 1) of a certain event. Such detection ability of a WSN is crucial for various applications. As an example, in a surveillance scenario the presence or absence of a target is usually determined, before attributes such as its position or velocity are estimated [37].
In distributed detection, by taking advantage of the limited onboard signal pro-cessing capabilities of sensors, the measurements are first preprocessed and a quantized version of the decision statistic is sent to the fusion center. For binary quantization and under different performance criteria (Bayes, Neyman-Pearson (NP)), the design of the optimal fusion rule is relatively straightforward but the evaluation of the decision thresholds at peripheral sensors is more complicated as a result of the distributed na-ture of the WSN. Therefore, obtaining local sensor decision rules is a major issue in the distributed detection problem [3].
For a given number of sensors and under the assumption of conditionally inde-pendent observations, the optimal decision rule at each sensor reduces to a likelihood ratio test (LRT) [3] for both Bayesian and NP criteria and different decision fusion
topologies such as parallel or serial. In parallel decision fusion, each sensor sends its decision directly to the fusion center whereas in serial decision fusion, all the sensors are connected in series. The routing path defines how these sensors are inter-connected and in this work we assume that it is known in advance. In the serial case, we assume that each sensor generates its decision by combining the decision coming from its pre-decessor with its own measurement. Then, the decision of the last sensor on the path is accepted as the final inference. Under decision fusion schemes for both topologies, the LRTs at each sensor are coupled with other sensor decisions and the fusion rule. Optimal values of the local sensor thresholds are typically found using Person by Per-son Optimization (PBPO) [3], where each sensor threshold is optimized iteratively by assuming a fixed fusion rule and decision rules at the other sensors. In the asymptotic regime where the number of sensors is very large, an identical decision rule for all the sensors is asymptotically optimal [36]. This result simplifies the design of decision rules considerably.
In this chapter, we assume ideal channels between the sensors and the fusion cen-ter (for recent work involving non-ideal channels, see [39], [6], [40]). Under the NP criterion and considering fading channels between sensors and the fusion center, an exhaustive search has been employed in [6] over all threshold selections to determine their optimal values. Computational complexity of such an approach increases expo-nentially with the number of local sensors and this approach for finding the optimal sensor thresholds is practical only with relatively few sensors. We assume that each sensor arrives at a binary decision about the event by comparing its decision statistic with a threshold. If the sensor decides positively about the presence of the event, it transmits one bit, otherwise it stays silent. To ensure perfect communication, each sensor decision should be transmitted with sufficient energy which is a function of the distance between the sensor and the fusion center [64]. Therefore, the thresholds of local sensors not only determine the network’s probability of error, but also affect the total energy consumption.
A recent work [47] considers the design of local sensor decision rules that minimize the probability of error subject to a transmission rate constraint for each sensor. Under
conditionally independent observations, a constrained minimization problem is defined and the optimal thresholds are obtained using the well known PBPO procedure. Al-though conditional independence assumption simplifies the derivation of decision rules, it may not be valid in many realistic cases such as when the location of the event isn’t known exactly. If the location of the event can only be described in terms of its probability density function, the received sensor decisions are no longer conditionally independent because of the unknown event location. Then the optimality of LRTs for local sensor decision making fails and the derivation of optimal sensor decision rules becomes complicated. In this chapter, we consider the case where the event has an isotropic signal emission with path loss [65], [66]. Then in the presence of the event, each sensor’s measurement depends on the distance between the sensor and the event location. Each noisy sensor measurement then follows the same probability distribution with different means as long as the measurement noise is independent and identically distributed across sensors. The sensors in proximity of the event decide more likely to decide on the presence of the event. In other words, an isotropic signal source for the event implies a high degree of spatial correlation. A related work [67] proposes a collab-orative detection scheme where a sensor close to the signal source requests collaboration and receives the decisions of the Kmax sensors within its neighborhood. The authors
showed that increasing Kmax, namely including more sensors to the collaboration that
are located far from the event degrades the detection performance considerably. Sensor network design usually involves simultaneous consideration of multiple con-flicting objectives [6], [68], such as maximizing the lifetime of the network or maximiz-ing the detection capability, while minimizmaximiz-ing the transmission costs. In a conventional WSN setting, one of the desired objectives is optimized while treating others as con-straints of the problem or the problem is converted into a single objective problem by assigning weights to each objective function. In the constrained minimization case, one single solution is obtained based on available resource limitations and the solution has to be reevaluated for each time when the amount of resource has been changed. In the weighted sum approach, relative weights of the objectives are usually not known or dif-ficult to determine. These drawbacks can be overcome via multi-objective optimization
methods [25], [26], [27], [28], [29], [30], [31] which optimize all the objectives simulta-neously and generate a set of solutions at the same time reflecting different trade-offs between the objectives. Multiobjective optimization has recently been introduced for WSN design [69] where the mobile agent routing and sensor placement problems and the tradeoff solutions between the desired objectives were determined through the use of multi-objective optimization based on evolutionary algorithms.
In this chapter, we study the event detection problem for sensor networks under isotropic signal emission and the event location is only known in terms of its proba-bility density function. Also, we assume that sensors employ the on-off keying scheme where they send one bit data to the fusion center only if they decide on the presence of the event. Then, sensor decision thresholds not only determine the probability of error but also determine the total energy consumption of the network. So, instead of having a single solution that minimizes the probability of error of the network, by using the multi-objective optimization approach, we seek several sensor threshold sets which deliver significant energy saving as compared to the energy consumption of the minimum probability of error solution without sacrificing probability of error too much. Thus, we are able to obtain a set of solutions which provide tradeoffs between energy consumption and probability of error performance.
Hence, we formulate a multi-objective optimization problem (MOP) with two objectives, minimizing the probability of error at the fusion center (global probabil-ity of error) Pe and minimizing the total network energy consumption (global energy
consumption) ET where the sensor decision thresholds are selected as the variables
of the MOP. We solve the MOP and generate the Pareto optimal solutions between these two conflicting objectives through Normal Boundary Intersection (NBI) [25] and Non-Dominating Sorting Genetic Algorithm II (NSGA - II) [28]. In this chapter, we first study the problem for parallel decision fusion where each sensor performed binary quantization by comparing its measurement with its threshold. We then compare the results of parallel decision fusion with serial decision fusion. In the serial case, it is hard to evaluate the optimal decision rule of each sensor since the event location is known only in terms of its probability density function. Simulation results show that when
each sensor makes its decision based on the decision of its predecessor and its own ob-servation, the performance is poor if the sensor is very far away from the event location. For this reason, motivated by the counting rule considered in [49], we use a heuristic decision rule at each sensor. Our decision statistic used for the serial case is the aggre-gation of sensor decisions from all the previous sensors and its own observation. In this work, we also compare the multi-objective optimization methods NBI and NSGA-II in detail by using the performance metrics, generational distance, domination and spacing metrics described in [29]. Finally, we compare the performance of the network both for different and identical sensor thresholds employed at each sensor.
The rest of the chapter is organized as follows. In Section 2.1, we state the WSN assumptions and describe each objective function under both parallel and serial decision fusion schemes. In Section 2.2, we review the fundamentals of MOP, describe NBI and NSGA-II methods. In Section 2.3, we present our simulation results and finally devote Section 2.4 to discussion of the results.
2.1 Problem Definition
In this section, we first state the wireless sensor network assumptions, then we define the mathematical models for both objective functions for parallel and serial decision fusion topologies.
2.1.1 Wireless Sensor Network Model and Statement of the MOP
A representative wireless sensor network consisting of N sensors, {sk, i = 1, 2, .., N }
with parallel decision fusion is shown in Figure 2.1. The distances between sk and the
fusion center and the event location (x, y) are denoted as df,k and dk respectively. We
assume the event location to be a random variable with an associated prior probability density function (pdf) and, therefore, dk is a random variable.
Specifically, we assume that the location of the event is uniformly distributed with joint pdf,
p(x, y) = 1
0 ≤ x ≤ A, 0 ≤ y ≤ B
where the region of interest (ROI) is an area of size A×B. Other pdfs can be employed in a similar manner. The average distance of sk located at (xk, yk) to the event location
(x, y) is then expressed as,
¯ dk = Z A 0 Z B 0 p (x − xk)2+ (y − yk)2p(x, y)dydx (2.2)
Suppose that a signal that follows the power attenuation model such as an acoustic signal is radiated from an event source with energy P0 [65] and sensors sk, i = 1, 2, .., N
are deployed at positions (xk, yk), i = 1, 2, .., N . Then, the received energy (ek) observed
at sk is,
Figure 2.1: Wireless Sensor Network Model with Parallel Decision Fusion
ek(xk, yk, x, y) = P0 dk ≤ d0 P0 ³ d0 dk ´n otherwise (2.3)
where n is the signal decay exponent and d0 is the reference distance where we select
d0 = 1m. When n = 2, the energy of the event decays at a rate inversely proportional to
the square of the distance dk =
p
(x − xk)2+ (y − yk)2. Then, under each hypothesis,
the received measurement of each sensor (zk) can be expressed as,
zk = nk, under H0 (2.4)
zk=
p
ek(xk, yk, x, y) + nk, under H1
where nk is the measurement noise that follows normal distribution at each sensor and
it is assumed to be independent across the sensors. zkthen follows a normal distribution
with parameters, zk∼ N(0, σ2) under H 0 N(pek(xk, yk, x, y), σ2) under H1 (2.5)
Throughout the chapter, we assume that the noise variance, σ2 is unity. When z k
exceeds a certain threshold denoted as tk, sensor sktransmits a one bit decision (Dk= 1)
to the fusion center. Otherwise, it does not transmit anything.
The functions global probability of error Pe and global energy consumption ET
are functions of the local sensor thresholds tk and constitute the objective functions of
the MOP. The MOP considered here is formulated as follows,
min
t1,t2,...,tN
{Pe(t1, t2, ..., tN), ET(t1, t2, ..., tN)}, (2.6)
tmin ≤ tk≤ tmax i ∈ {1, 2, ..., N }.
We first solve the above problem for N nonidentical decision thresholds {t1, t2, ..., tN}
employed at each sensor. We also compare the performance of nonidentical decision thresholds with identical decision threshold at each sensor {t = t1 = ... = tN} via
simulation.
In the next subsections, we derive the objective functions for the global probability of error and the global energy consumption under parallel and serial decision fusion
